The word 'nominative' is from the Latin
nomen '
name', because it is the form in which you name something in isolation: that is, it is the
citation form or
dictionary form. You say "the Latin for 'master' is
dominus".
This is also the case used in certain functions in a sentence, as described above. The most basic one is the subject of a sentence: in 'John kisses Mary' or 'John sleeps' or 'John is hungry', John is nominative.
Now in English nouns have no case marking except for the genitive in 'John's book', so John has the same form in 'Mary kisses John' as in 'John kisses Mary'. If we're talking about morphology, the actual shapes of words, we can say they're both the same, nominative. But English pronouns do show a difference between subject and object forms: 'He kisses Mary' versus 'Mary kisses him'. The nominative pronouns include he/she/I/we/they, while the accusative ones are him/her/me/us/them. The pronouns it/you are like nouns in not showing a distinction. But we can say that the underlying grammar of English does have this distinction, but not all words show it. So, in this underlying syntactic sense, John is nominative in 'John kisses Mary' but accusative in 'Mary kisses John'.
It is very rare to find any discrepancy between the subject form and the citation form, but at least logically they are distinct. I'll now go over some language situations where the two kinds of use might be considered different.
In Polynesian languages the tendency is to cite words with a particle o, whose exact meaning is hard to describe: "subject" isn't quite right. So o ika 'fish', where we would actually find ika in a dictionary.
In Basque the subject (of an intransitive sentence) usually has an article -a attached (if it has no other more specific article). So 'woman' is emakume but 'the woman is old' is emakumea zaharra da.
Neither instance is important, because the extra element can be compared to the English article 'the'.
When applied further afield than Latin, Greek, and related European languages, the nominative is specifically the case of the subject of an intransitive sentence, that is one with no direct object. Languages where the subject of a transitive sentence is differently marked are called ergative. The transitive subject case is also called ergative; and the nominative is used for the direct object as well as the intransitive subject. In this situation some linguists prefer to use the term absolutive (or absolute) instead of nominative, which they restrict to languages that work on a nominative-accusative system. See typology for more explanation.
A few languages (Georgian is one) vary between nominative-accusative and ergative-absolutive depending on tense.
The nominative is sometimes called subjective, especially in grammars of the English language. I can't see any good reason for this.